"More importantly, they have surrendered to Obamacare — the biggest job killer in America," the group declared in an email statement, according to Politico.

The group's email blast came as Cruz launched into a nearly day-long mock filibuster aimed at preventing a vote from taking place in the Senate on a House-passed stopgap spending measure that would continue to fund the government but eliminate funding to implement the healthcare reform law.

The Democratic-controlled Senate is almost certain to reject the measure and send it back to the House with Obamacare funding restored. But many Senate Republicans worry that Cruz's attempt to delay the vote could lead to a government shutdown for which the party would ultimately be blamed.

Both McConnell and Cornyn are opposed to the filibuster tactics being pursued by Cruz, but their spokesmen insisted Tuesday that they remain opposed to any funding for Obamacare.

"Senator McConnell supports the House Republicans' bill and will not vote to block it, since it defunds Obamacare and funds the government without increasing spending by a penny," a McConnell spokesman said in a statement, Politico reported. "He will also vote against any amendment that attempts to add Obamacare funding back into the House Republicans' bill."

Likewise, a Cornyn spokesperson told The Huffington Post that her boss "will not block a bill that defunds Obamacare."

Still, the Senate Conservatives Fund accused McConnell and Cornyn of trying to have it both ways.

"We knew Mitch McConnell and John Cornyn weren't with us when they voted to fund Obamacare earlier this year," the group said. "But then, under pressure from grassroots, they said they supported the effort to defund it. They even ran political ads in their home states to make voters think they were on their side."

The Senate Conservatives Fund has attacked McConnell before but has stopped short of officially declaring its support for his tea party-affiliated primary opponent, Republican Matt Bevin. The group, however, has put up a poll on its website asking supporters if it should endorse the tea party challenger.